In this talk, I will present some recent results regarding isometric embedding. I first review some conclusions on isometric immersions and isometric extension and related problem. Then I will show our global extensions of the celebrated Nash-Kuiper theorem for isometric immersions of compact manifolds with optimal H\"older exponent. In particular for the Weyl problem of isometrically embedding a convex compact surface in 3-space, we show that the Nash-Kuiper non-rigidity prevails upto exponent . This extends previous results on embedding 2-discs as well as higher dimensional analogues. The presented results are joint work with my mentor Prof. Dr. Szekelyhidi in Leipzig University.